14

The lattice Boltzmann method for fluid dynamics

14.1 Introduction

Flow problems are widely studied in engineering because of their relevance to
industrial processes and environmental problems. Such problems belong to the
realm of macroscopic phenomena which are formulated in terms of one or more,
possibly nonlinear, partial differential equations. If there is no possibility of exploit-
ing symmetry, allowing for separation of variables, these equations are usually
solved using finite element or finite difference methods.
The standard problem is the flow of a one-component, isotropic nonpolar liquid,
which is described by the Navier–Stokes equations. These equations are based on
mass and momentum conservation, and on the assumption of isotropic relaxation
towards equilibrium. Finite element methods have been described in Chapter 13
for elasticity; the methods described there may be extended and adapted to develop
codes for computational fluid dynamics (CFD), which are widely used by engineers.
Such an extension is beyond the scope of this book.
A finite element solution of the Navier–Stokes equations may sometimes become
cumbersome when the boundary conditions become exceptionally complicated, as
is the case with flow through porous media where the pore diameter becomes
very small (and the number of pores very large). Other cases where finite element
methods run into problems are multiphase or binary systems, where two different
substances or phases exist in separate regions of space. These regions change their
shape and size in the course of time. Usually, the finite element points are taken
on the system boundaries, but that implies complicated bookkeeping, in particular
when the topology of the regions changes, as is the case in coalescence of droplets.
A possible solution to these problems characterized by difficult topologies is to
use regular, structured grids, and let the interfaces move over the grid. In that case,
we need special couplings for nearest neighbour grid points that are separated by the
interface (such as a phase boundary). This ‘immersed interface’ method has been

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